In Sec.  of the paper we turn to our major topic, that
is, the ST of FBM. FBM is a non-Markovian process that can generate
all modes of anomalous diffusion, from sub- to normal, to
superdiffusion. This property of FBM makes it a versatile and nowadays
widely used model for numerous experimental observations of anomalous
diffusion in nature and laboratories [1]}. Examples include
the motion of tracers in viscoelastic media [2]}, crowded
in-vitro environments [3]}, [4]}, [5]}, living cells [6]}, [7]}
and intracellular media [8]}. Given its quite universal
applicability, the investigation of ST concepts for FBM systems is
both timely and relevant. Our goal here is to unravel the challenges
implied by the non-Markovianity and the absence of the FDR for
definition of heat production, entropy production, and the related
IFR. To this end, we employ a fractional differential approach and a
perturbation expansion. As a main result, we provide explicit
expressions and an interpretation for the leading corrections induced
by non-Markovianity to the usual temperature and heat.
